Some Fun with GNUPLOT

Introduction

This page contains a couple of cool graphs
done with GNUPLOT. GNUPLOT is great and free!
GNUPLOT is available for download
here.

The GNUPLOT commands used to generate each
graph are listed after each figure. Note that
in a couple of cases the formula lines are very long.
This is indicated with backslash ("\") characters.
In GNUPLOT, the backslash means that the following
line is a continuation of the current one. You must
either enter the commands exactly as shown (with the "\"
followed by a new line) or combine them into a single line
omitting the "\" characters.

Gif files were produced with the following commands:

set terminal gif
set output "out.gif"
replot

I.e., if you type in the GNUPLOT commands and can see
the a figure on your screen, these commands will save
the image as a .gif file.

Mobius Surface

The Mobius surface (or Mobius "Strip") is a one-sided object, with
only one edge. It may appear at first glance to have two sides,
however it is possible for one to walk from one side to the
other without having to cross the edge - because of the twist.

Klein Bottle

The Klein bottle, similar to the Mobius surface above,
is one-sided; however unlike the Mobius surface, it has no edges. One
way to think of it, is as two Mobius strips joined at their edges
(the corollary is that if you cut a Klein Bottle in half along
its length you get two Mobius strips).

A Klein bottle can only really exist in four (or higher) dimentions; in
three dimensions the neck of the bottle bends around and must pierce the
side of the bottle. In four dimentions the klein bottle does not intersect
itself. So what you see below is really a three-dimensional immersion
of a four-dimentional object (although it's really a two-dimensional
projection of a three dimensional immersion...).

Think of a regular wine bottle with a cork - if you are outside, you
cannot get inside without getting past the cork. A bottle is topologically
isomorphic to a disc - there's a top and a bottom, separated by an edge.
The cork serves the function of blocking the edge of the bottle.

If one were to stick
a cork in the neck of a Klein bottle (in the bottom of the figure below),
it is still possible to get inside without having to get past the cork.
One simply follows the side of the bottle up, around, down to the neck,
and through the side, to end-up
inside the bottle without ever having to squeeze past the cork. You
would have to (along with the neck) squeeze through the side of the bottle
without breaking it, but in four dimensions this is easy to do.

A fun site with lots of information, and with real three-dimensional
immersions of Klein bottles for sale is
available here.
A more mathematically rigorous site
appears here.
(Note that I am not associated with either of these two sites.)

"Figure-8" parameterisation...

This Klein surface can be constructed by rotating a figure eight
about its axis while inserting a twist in it.
This results
in the figure appearing below. While (in four dimensions) this is
topologically isomorphic to the Klein bottle above it's not as
aesthetically appealing as the previous figure.